We extend the argument of [1]}. For the first statement, as \(p\)  is \(U\) -invariant and restricts to the identity on \(Z_{\min }\) , if \(uz \in Z_{\min }\) , we see \(p(uz) = uz\)  and also \(p(uz) = z\) , and so \(u\in \operatorname{Stab}_U(z)\) . For the second statement, any \(h \in H\)  can be written as \(h=ul\)  for \(u\in U\)  and \(l\in L\) . If \(hz= z\)  then \(uz = l^{-1}z \in Z_{\min }\)  since \(L\)  preserves \(Z_{\min }\)  as the grading 1-PS \(\lambda \)  is central in \(L\) . Thus any \(h\in \operatorname{Stab}_H(z)\)  can be written as a product of \(ul\)  with both elements fixing \(z\) . Since \(\operatorname{Stab}_U(z)\cap \operatorname{Stab}_L(z)= \lbrace e\rbrace \)  the result follows.
